A 0.20 m uniform bar has a mass of 0.75 kg and is released from rest in the vertical position, as the drawing shows. The spring is initially unstrained and has a spring constant of 25.0 N/m. Find the tangential speed with which end A strikes the horizontal surface.

To find the tangential speed with which end A strikes the horizontal surface, we can use the principle of conservation of mechanical energy. We need to calculate the potential energy at the center of mass, gravitational potential energy at height h, and the kinetic energy when the bar reaches the horizontal position.

1. Calculate the potential energy at the center of mass:
The potential energy at the center of mass can be calculated as the gravitational potential energy of the entire bar. The formula for gravitational potential energy is given by U = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height.

In this case, since the bar is initially in the vertical position, the height h is the length of the bar divided by 2 (assuming the center of mass is at the center of the bar). So, h = 0.20 m / 2 = 0.10 m.

Using the formula U = mgh, where m = 0.75 kg and g = 9.8 m/s^2, we can calculate the potential energy at the center of mass.

U = (0.75 kg) * (9.8 m/s^2) * (0.10 m)
U = 0.735 J

2. Calculate the gravitational potential energy at height h:
The gravitational potential energy at height h can be calculated using the same formula as before. In this case, h is equal to the length of the bar, which is 0.20 m.

Using the formula U = mgh, where m = 0.75 kg and g = 9.8 m/s^2, we can calculate the gravitational potential energy at height h.

U = (0.75 kg) * (9.8 m/s^2) * (0.20 m)
U = 1.47 J

3. Calculate the total mechanical energy:
The total mechanical energy is the sum of the potential energy at the center of mass and the gravitational potential energy at height h.

Total mechanical energy = Potential energy at the center of mass + Gravitational potential energy at height h
Total mechanical energy = 0.735 J + 1.47 J
Total mechanical energy = 2.205 J

4. Calculate the kinetic energy at the horizontal position:
At the horizontal position, all potential energy is converted into kinetic energy. Therefore, the kinetic energy at the horizontal position is equal to the total mechanical energy.

Kinetic energy = Total mechanical energy = 2.205 J

Now, we can use the formula for kinetic energy to calculate the tangential speed (v) when the bar reaches the horizontal position. The formula for kinetic energy is given by KE = (1/2) * m * v^2, where KE is the kinetic energy, m is the mass, and v is the velocity.

Using the formula for kinetic energy, we can calculate the tangential speed (v).

2.205 J = (1/2) * 0.75 kg * v^2

Rearranging the equation and solving for v:

v^2 = (2 * 2.205 J) / (0.75 kg)
v^2 = 5.88 m^2/s^2
v = √(5.88 m^2/s^2)
v = 2.42 m/s

Therefore, the tangential speed with which end A strikes the horizontal surface is 2.42 m/s.